# The upper attic

From Cantor's Attic

Welcome to the upper attic, the transfinite realm of large cardinals, the higher infinite, carrying us upward from the merely inaccessible and indescribable to the subtle and endlessly extendible concepts beyond, towards the calamity of inconsistency.

- The Kunen inconsistency: Reinhardt cardinal, super Reinhardt cardinal, Berkeley cardinal
- Rank into rank cardinals $j:V_\lambda\to V_\lambda$, rank+1 into rank+1 cardinal $j:V_{\lambda+1}\to V_{\lambda+1}$, I0 cardinal $j:L(V_{\lambda+1})\to L(V_{\lambda+1})$
- The wholeness axiom
- almost n-huge cardinal, n-huge cardinal, (n+1)-superstrong cardinal, super-n-huge cardinal
- Vopěnka's principle, Vopěnka cardinal, Woodin for supercompactness cardinal, high-jump cardinal
- $\alpha$-extendible cardinal, extendible cardinal
- grand reflection cardinal
- $\lambda$-supercompact cardinal, supercompact cardinal
- PFA cardinal
- strongly compact cardinal
- nearly supercompact and nearly strongly compact cardinals
- indestructible weakly compact cardinal
- subcompact cardinal
- superstrong cardinal
- Shelah cardinal
- Woodin cardinal, the axiom of determinacy
- strong cardinal and the $\theta$-strong and hypermeasurability hierarchy
- tall cardinal
- $0^\dagger$, $j:L[U]\to L[U]$ cardinal
- Nontrivial Mitchell rank, $o(\kappa)=1$, $o(\kappa)=\kappa^{++}$
- weakly measurable cardinal, measurable cardinal
- virtually Ramsey cardinal, Ramsey cardinal, strongly Ramsey cardinal
- Rowbottom cardinal
- Jónsson cardinal
- $\omega_1$-Erdős cardinal and $\gamma$-Erdős cardinals for uncountable $\gamma$
- $0^\sharp$, $j:L\to L$ cardinal
- Erdős cardinal, and the $\alpha$-Erdős hierarchy for countable $\alpha$
- $1$-iterable cardinal, and the $\alpha$-iterable cardinals hierarchy for $1\leq \alpha\leq \omega_1$
- remarkable cardinal
- weakly ineffable cardinal, ineffable cardinal, and the $n$-ineffable cardinals hierarchy; completely ineffable cardinal
- subtle cardinal
- ethereal cardinal
- superstrongly unfoldable cardinal, strongly uplifting cardinal
- weakly superstrong cardinal
- unfoldable cardinal, strongly unfoldable cardinal
- indescribable cardinal, totally indescribable cardinal
- weakly compact cardinal
- hyper-Mahlo cardinals
- Mahlo cardinal, $1$-Mahlo, the $\alpha$-Mahlo hierarchy
- psuedo uplifting cardinal, uplifting cardinal
- ORD is Mahlo
- $\Sigma_2$-reflecting, $\Sigma_n$-reflecting and reflecting cardinals
- $1$-inaccessible, the $\alpha$-inaccessible hierarchy and hyper-inaccessible cardinals
- Grothendieck universe axiom, equivalent to the existence of a proper class of inaccessible cardinals
- weakly inaccessible cardinal, (strongly) inaccessible cardinal,
- Kelly-Morse set theory
- worldly cardinal and the $\alpha$-wordly hierarchy, hyper-worldly cardinal
- the transitive model universe axiom
- Transitive ZFC model
- the minimal transitive model
- Con(ZFC) and $\text{Con}^\alpha(\text{ZFC})$, the iterated consistency hierarchy

- down to the middle attic